uni_notes/DAS/Set Theory.md

A set is a collection of unordered elements. A set cannot contain duplicates.

Notation

Set Notation

Declaration of a set A with elements a, b, c:

A := {a, b, c}

Cardinality

Amount of Elements in a set A Notation: |A|

A := {1, 2, 3, 4} \
|A| = 4

Well-Known Sets

• Empty Set: emptyset = {}
• Natural Numbers: N = {1, 2, 3, ...}
• Integers: ZZ = {-2, -1, 0, 1, 2}
• Rational Numbers: QQ = {1/2, 22/7 }
• Real Numbers: RR = {1, pi, sqrt(2)}
• Complex Numbers: CC = {i, pi, 1, sqrt(-1)}

Set-Builder Notation

Common form of notation to create sets without explicitly specifying elements.

A := {x in N | 0 <= x <= 5} \
A = {1, 2, 3, 4, 5}

Member of

Denote whether x is an element of the set A Notation: x in A Negation: x in.not A

Subsets

Type	Explanation	Notation
Subset	Every element of A is in B	A subset B
Subset or equal to	Every element of A is in B, or they are the exactly same set	A subset.eq B
Proper subset	Every element of A is in B, but A is definitely smaller than B	$A subset.sq B$
Superset	A contains everything that is in B	A supset B
Superset or equal to	A contains everything that is in B, or they are identical	A supset.eq B

Operations

Union

Notation: A union B Definition: all elements from both sets without adding duplicates

A := {1, 2, 3}\ B := {3, 4, 5}\ A union B = {1, 2, 3, 4, 5}

Intersection

Notation:$A inter B$ Definition: all elements contained in both sets

A := {1, 2, 3} \
B := {2, 3, 4} \
A inter B = {2, 3}

Difference

Notation: A backslash B Definition: all elements in A that are not in $B$

A := {1, 2, 3} \
B := {3, 4, 5} \
A backslash B = {1, 2}

Symmetric Difference

Notation: A Delta B Definition: all elements only in A or only in $B$

A := {1, 2, 3} \
B := {2, 3, 4} \
A Delta B = {1, 4}

Cartesian Product

Notation: A times B Definition: all pairs of all elements in A and B

A := {1, 2} \
B := {3, 4, 5} \
A times B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}

Powerset

Notation: cal(P)(A) Definition: all possible Subsets of A

A := {1, 2, 3} \
cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

Relations

Relation	Explanation	Example
transitive	"chain reaction", a information about a in relation to c can be inferred from the relations a -> b and b -> c	a < b, b < c => a < c
reflexive	every element is related to itself with the given relation	a <= a, 5 = 5
anti-reflexive	every element is NOT related to itself in the given relation	a < a